In the main text, we consider only the lattice relaxation proposed in Ref. [1]} and [2]}, in which the lattice distortion is dominated by the solenoid part. However, in this section, for completeness, we consider a more general \(U_j^{\parallel }\) , whose irrotational part may also be important, and only in the last step, we set \(\nabla \cdot \delta U = 0\) .
As mentioned in the main text, up to the second order of the derivatives, the intra-layer part is
\(H_{intra} & = H^{(0)}_{intra} + \delta H_{intra} \\H^{(0)}_{intra} & = \int {\rm d}^2x\ \sum _j \sum _{S S^{\prime }}\Psi _{j, S}^{\dagger }(x) \left\lbrace  \mu \delta _{SS^{\prime }} + v_F \bar{\sigma }_{S S^{\prime }} \cdot \left( p^{(j)} + \gamma \mathcal {A}^{(j)} \right) + \alpha _{dp} \phi ^{(j)} \delta _{SS^{\prime }} + \beta _0 p^2 \delta _{S S^{\prime }} -\frac{C_0}{2} \left( p \cdot \mathcal {A}(x) + \mathcal {A}(x) \cdot p \right) \delta _{S S^{\prime }} \right. \nonumber \\& \left. + \beta _1 \left( ( p_x^2 - p_y^2) \sigma _1 + 2 p_x p_y \sigma _2 \right)_{S S^{\prime }} \pm \frac{1}{2}\left( p_{\mu } \xi _{\mu ,SS^{\prime }}(x) + \xi _{\mu ,SS^{\prime }}(x) p_{\mu } \right) + 2D_0 \left\lbrace  \phi ^{(j)}, \bar{\sigma }_{S S^{\prime }} \cdot p \right\rbrace  \right\rbrace  \Psi _{j, S}(x) \\\delta H_{intra} & = \int {\rm d}^2x\ \sum _j \sum _{S S^{\prime }}\Psi _{j, S}^{\dagger }(x) \left\lbrace  \sum _{\mu \nu } \left[ C_1 \left( (\nabla \cdot \delta U)^2 + (\partial _{\mu } \delta U_{\nu }) (\partial _{\nu } \delta U_{\mu }) \right) + C_2 (\partial _{\mu } \delta U_{\nu }) (\partial _{\mu } \delta U_{\nu }) \right]\delta _{S S^{\prime }} \right. \nonumber \\& \left. + \alpha \left(-\frac{\theta }{2} (\nabla \times \delta U)_z - \frac{1}{2}\left( \nabla \cdot \delta U \right)^2 \right) \delta _{S S^{\prime }} + \zeta _{S S^{\prime }}(x) \right\rbrace  \Psi _{j, S^{\prime }}(x) \ . \) 
